SOLUTION: Prove that: {{{(tan^2(x)^""+1)cos(2x)}}}{{{""=""}}}{{{2-sec^2(x)}}}

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Question 1053683: Prove that:
%28tan%5E2%28x%29%5E%22%22%2B1%29cos%282x%29%22%22=%22%222-sec%5E2%28x%29
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
%28tan%5E2%28x%29%5E%22%22%2B1%29cos%282x%29%22%22=%22%222-sec%5E2%28x%29

We work with the left side only:

%28tan%5E2%28x%29%5E%22%22%2B1%29cos%282x%29%22%22=%22%22

Use the commutative principle to write tan%5E2%28x%29%2B1 as 1%2Btan%5E2%28x%29

%281%2Btan%5E2%28x%29%29cos%282x%29%22%22=%22%22

We use the two identities:  1%2Btan%5E2%28theta%29=sec%5E2%28theta%29
                            cos%282theta%29+=+2cos%5E2%28theta%29-1

sec%5E2%28x%29%282cos%5E2%28x%29%5E%22%22-1%29%22%22=%22%22

We use the identity sec%28theta%29=1%2Fcos%28theta%29

%281%2Fcos%28x%29%29%5E2%282cos%5E2%28x%29%5E%22%22-1%29%22%22=%22%22

Use principle of exponents:

%281%5E2%2Fcos%5E2%28x%29%29%282cos%5E2%28x%29%5E%22%22-1%29%22%22=%22%22

Replace 1² by 1

%281%2Fcos%5E2%28x%29%29%282cos%5E2%28x%29%5E%22%22-1%29%22%22=%22%22

We use the distributive principle:

%281%2Fcos%5E2%28x%29%29%282cos%5E2%28x%29%5E%22%22%29-1%281%2Fcos%28x%29%29%29%22%22=%22%22

Cancel the cos²(x)'s

%281%2Fcross%28cos%5E2%28x%29%29%29%282cross%28cos%5E2%28x%29%29%29-1%281%2Fcos%28x%29%29%29%22%22=%22%22 

 2-1%2Fcos%5E2%28x%29

Use the identity 1%2Fcos%28theta%29=sec%28theta%29

2-sec%5E2%28x%29

Edwin